Varieties with definable factor congruences

Abstract

Abstract. We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are definable in a first-orderlogic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that V has DFC if and only if V has 0 & 1 and Boolean Factor Congruences. We also obtain an explicit first order definition Φ of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DFC, in such a manner it is preserved by taking direct products and direct factors. The main tool is the use of central elements, which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice.

Directly Indecomposables in Semidegenerate Varieties of Connected po-Groupoids

“…In a joint work with D. Vaggione [7] it is proved that semidegenerate varieties with the SRP have definable factor congruences and if the similarity type is finite, directly indecomposables are axiomatizable by a set of first-order sentences. Our main result is an application of [7] and a result of R. Willard [11]. Theorem 2 Let V be a semidegenerate variety of connected po-groupoids over a finite language.…”

“…Also, in [7,Section 6] it is shown that semidegeneracy by itself does not ensure definability of directly indecomposables. By considering in this last case a trivial (antichain) po-groupoid structure (for instance, defining x · y := y for all x, y) we deduce that an arbitrary semidegenerate variety of po-groupoids may not have a first-order-axiomatizable class of indecomposables; hence we cannot drop connectedness.…”

Coextensive varieties via central elements

“…Within the context of varieties in which the universal congruence of each algebra is compact, in [25] Vaggione introduces the concept of central element. In [19] it is proved that under the aforementioned condition, varieties with Boolean factor congruences are equivalent to varieties in which factor congruences are definable by a first order formula. This paper is motivated by the fact that some algebraic categories studied by the author (see [7] and [29]) were coextensive varieties and the proof of this fact always seemed to be related precisely with those elements which concentrate all the information about direct product decompositions.…”

The Fundamental Theorem of Central Element Theory

“…In this note we give a short proof of the fundamental theorem of central element theory [7,Theorem 1]. The proof given in [7] is constructive and very involved and relies strongly on the fact that the class be a variety (i.e. equationally definable class of algebras).…”